Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

Guanghui Hu; Yavar Kian

doi:10.3934/ipi.2020022

Article Contents

2020, Volume 14, Issue 3: 463-487. Doi: 10.3934/ipi.2020022

This issue Previous Article Regularization of inverse problems via box constrained minimization Next Article Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency

Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

Guanghui Hu^1, , and
Yavar Kian^2,

1.
School of Mathematical Sciences, Nankai University, Tianjin, China, 300071
2.
Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

^* Corresponding author
^* Corresponding author

Received: June 30, 2019

Revised: November 30, 2019

Published: March 2020

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Abstract

This paper is concerned with inverse source problems for the time-dependent Lamé system in an unbounded domain corresponding to either the exterior of a bounded cavity or the full space $ {\mathbb{R}}^3 $. If the time and spatial variables of the source term can be separated with compact support, we prove that the vector valued spatial source term can be uniquely determined by boundary Dirichlet data in the exterior of a given cavity. Uniqueness and stability for recovering some class of time-dependent source terms are also obtained by using, respectively, partial and full boundary data.

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Mathematics Subject Classification: Primary: 65M32, 35R30; Secondary: 74B05.

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Figure 1. Radiation of a source in an inhomogeneous isotropic elastic medium in the exterior of a cavity. Suppose that the cavity $ D $ is known. The inverse problem is to determine the source term from the data measured on $ \partial B_R = \{x\in {\mathbb{R}}^3: |x| = R\} $

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Figure 2. Suppose that the data are collected on $ \omega $ and on $ \partial\Omega $. The inverse problem is to determine the value of $ g $ on $ \Omega $

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